Keywords: discrete crack models, cohesive fracture, thermal loads, adaptive remeshing, bimaterial sample, PMMA.

The paper presents a fully-coupled numerical model for the analysis of fracture initiation and propagation in a two dimensional elastic medium driven by transient mechanical forces and/or thermal field. The problem of fracture is formulated within the frame of the cohesive model, which is very realistic for geomaterials, but is also widely applied to other materials [1]. The medium is assumed as nonhomogeneous and the fracture/s can enucleate and propagate according to the maximum tensile stress criterion. Crack path is unknown a priori and may be influenced by the presence of material interfaces. At the same time, the temperature field can evolve in space and time according to energy balance equation.

Governing equations are obtained within the framework of the phenomenological theory [2], under small displacements and displacement gradients assumptions. The model requires the local linear momentum balance equation, where inertial forces are present. This equation is supplemented by the natural boundary condition, containing applied tractions and cohesive forces in the process zone. Further, the model involves the local energy balance equation (first law of thermodynamics), which states the conservation of the specific internal energy, in the presence of mechanical interactions, distributed internal heat sources and heat flux. When mechanical terms can be neglected, internal energy depends on temperature only. This is the case of the studied application.

The assumed constitutive relationships are:

• Green-elastic material. The mechanical behaviour depends on the effective strain and temperature;
• Cohesive forces in accordance with Barenblatt-Dugdale model for mode I fracture;
• Cohesive forces following Camacho-Ortiz [5] in the case of mixed mode;
• Fourier's law for heat flux.

The finite element method is assumed for spatial discretisation, whereas generalized trapezoidal rule is used for time integration. Monolithic solution procedures are used for the resulting algebraic system of equations, i.e. the governing equations are solved simultaneously to obtain the displacement and temperature fields together with the fracture path. The numerical solution is limited to a 2-D context. No special approximations are used to represent the field singularities: linear interpolations are assumed for both field variables and the solution is controlled and improved by using an error measure together with an a posteriori refinement technique. Spatial discretization is continuously updated as the phenomenon evolves and the domain of definition changes. An efficient mesh generator [3,4] is used to this scope and the initial mesh can also be defined without special care. Cracks may enucleate everywhere depending only on the stress field and propagate along paths and with a velocity of the tip that is unknown a priori. The determination of the crack path and the velocity of the tip represent an important part of the solution, as the temperature and stress fields and allows for correct updating of the domain, boundaries and related conditions in the following steps. It should be noted that the topology of the domain and boundary change with the evolution of the fracture phenomenon. In particular, the fracture path, the position of the process zone and the cohesive forces are unknown and must be regarded as products of the mechanical analysis.

The experimental results of a three-point bending test performed on a bimaterial specimen (aluminium and PMMA) subjected to thermo-mechanical loading [6] is used as a benchmark. Even though in the lack of some mechanical characteristics, numerical results compare satisfactorily as far as temperature field and crack path is concerned. The application reveals however the potentiality of the model which is capable to analyse the behaviour of the bonded interface.

1

Elices, M., Guinea, G.V., Gomez, J., Planas, J., The coesive zone model: advantages, limitations and challanges, Engineering Fracture Mechanics, 69, 137-163, 2002. doi:10.1016/S0013-7944(01)00083-2

2

Lemaitre, J., Chaboche, J.L., Mechanics of solids materials, Cambridge University Press, 1990.

3

Secchi S. and Simoni L., An improved procedure for 2-D unstructured Delaunay mesh generation, submitted for publication in Advances in Engineering Software.

4

Simoni L., Secchi S., Cohesive fracture mechanics for a multi-phase porous medium, in http://congress.cimne.upc.es/femclass42/html/papers.asp.

5

Camacho G.T., Ortiz M., Computational modelling of impact damage in brittle materials, Int. Journal of Solids and Structures, 33, 2899-2938, 1996. doi:10.1016/0020-7683(95)00255-3

6

Joon-Soo Bae, Sridhar Krishnaswamy, Subinterfacial cracks in bimaterial systems subjected to mechanical and thermal loading, Engineering Fracture Mechanics, 68, 1081-1094, 2001. doi:10.1016/S0013-7944(01)00005-4

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